Rydberg formula
/ RID-berg /
Long before anyone knew why, scientists noticed that the bright lines in hydrogen's spectrum were not scattered randomly — their wavelengths followed a tidy numerical pattern, as regular as notes on a scale. The Rydberg formula is the simple equation that captures that pattern, letting you calculate exactly where each spectral line falls using only whole numbers.
The Rydberg formula gives the wavelengths of light a hydrogen atom emits or absorbs. It says that one over the wavelength equals the Rydberg constant times the difference of one over n-lower squared minus one over n-upper squared, where n-lower and n-upper are the whole-number labels of the two energy levels involved. From just two integers, it predicts a precise colour of light.
It mattered because it was a quantitative bullseye that any theory of the atom had to hit — and Bohr's model, then quantum mechanics, did. The caveat is that the simple formula is exact only for hydrogen and other one-electron ions; for multi-electron atoms the clean integer pattern blurs, and corrections are needed.
For the red Balmer line, set the lower level to 2 and the upper to 3. The formula gives one over the wavelength as the Rydberg constant times (1/4 − 1/9), which works out to a wavelength of 656 nanometres — the exact red glow of a hydrogen lamp.
Two integers in, an exact spectral wavelength out.
Different choices of the lower level give named series: n-lower = 1 is the ultraviolet Lyman series, n-lower = 2 the visible Balmer series, n-lower = 3 the infrared Paschen series.